Particle Adhesion and Removal in Model Systems Part 1 .-Monodispersed Chromium Hydroxide on Glass BY JAN E. KOLAKOWSKI~ AND EGON MATIJEVIC" Institute of Colloid and Surface Science and Department of Chemistry, Clarkson College of Technology, Potsdam, New York 13676, U.S.A. Received 7th March, 1978 The interactions of monodispersed chromium(m) hydroxide particles of 0.28 pm diameter with glass were studied using the packed column technique. The particles were first adsorbed by passing the sol through a bed containing glass beads at pH - 3, at which conditions the particles and the beads are of opposite charge. Desorption was then studied as a function of pH by rinsing the column with solutions containing different electrolytes in varying concentrations. Optimum removal at low ionic strength occurred at pH 11 5, which is well above the point of zero charge of the colloidal chromium hydroxide particles.When the particles were adsorbed at pH 4 and - 6, respectively, the removal was substantially less under otherwise identical conditions. The addition of NaN03, NaF, Ca(NO& and C~(dipy)~(ClO& at the pH of optimum removal (11.5) of the rinse solution caused a decrease in desorption and, depending on ionic strength, eliminated altogether particle separation. The greater the counterion charge, the less salt was necessary to suppress desorption. These resuIts are best explained in terms of double layer interactions between the particles and the substrate and no indication of chemical bonding could be detected. The removal of adhered colloidal particles from substrates immersed in liquids or solutions is important in many processes, such as in the cleaning of soiled materials, in the elimination of oxide scale from metals, etc.Most of the work, both experi- mental and theoretical, involving fine particle adhesion is concerned with the de- position (or adsorption) of such particles rather than their removal. Particle deposition has been frequently studied with the rotating disc apparatus. The technique, the theory of which was given by Levich,l was originally used in the investigations of the transport of ions.2 The first application of the rotating disc principle to particle adhesion was given by Marshall and Kit~hener,~ who followed the deposition of carbon black from dilute aqueous suspensions on glass and other substrates.Later, Hull and Kitchener deposited polystyrene latex onto a glass disc coated with either cationic or anionic polymers, and Clint et aL5 examined the adhesion of polystyrene latex on films of polystyrene cast onto glass discs. When the latex particles and the polymer were of opposite charge, the deposition rate followed the Levich equation but the existence of a potential energy barrier to adsorption required a modification of the theory. Using the same technique, Tewari and Camp- bell studied the adhesion of chromium hydroxide spheres and of rod-like P-FeOOH particles onto stainless steel discs, both in the presence and in the absence of a potential barrier. Clayfield and Lumb developed a packed column method to investigate the detach- ment of adhered colloidal particles.The technique was applied to the system carbon black/metal surface in non-aqueous and aqueous surfactant solutions,* and to deposition of monodispersed latexes on glass. p This work is part of a ms. thesis by J. E. K., Clarkson College, 1977. 1-3 6566 PARTICLE ADHESION AND REMOVAL Particle removal may be difficult to interpret because of surface irregularities, the possibility of chemical bond formation, mechanical deformation, hydrodynamic effects, and other phenomena that can occur between bodies in contact. The problem is considerably simplified if only electrical double layer forces and/or diffusion play a role. In this work a well defined model system has been studied. The removal of mono- dispersed spherical colloidal chromium hydroxide particles deposited on glass beads in a packed column was measured as a function of pH, ionic strength and counter- ion charge.The chosen system is well suited for this type of investigation because the particle charge can be readily changed and even reversed by varying the pH. Furthermore, since the particles are uniform in size, their concentration is easily monitored. The results obtained are interpreted in terms of the existing theory of the electrical double layer. EXPERIMENTAL MATERIALS The glass beads were Ballotini type 0, of which > 85 % were in the form of true spheres. The fraction of a sample, passing a no. 230 and retained by a no. 325 A.S.T.M. sieve (resulting in beads with a diameter between 44 and 62pm), was collected and cleaned by stirring in aqua regia followed by a thorough rinse with doubly distilled water.The glass powder was used after being dried overnight in an oven at 95°C and then cooled in a vacuum desiccator. The preparation of the hydrous chromium(1Ir) oxide sols was described in detail else- where.l0 The sols were obtained by ageing a solution 4x mol dm-3 in CT(NO~)~ and 6x mol dm-3 in KzS04 at 75&3"C for 72 h. This gave a modal particie radius of 0.14pm. The sol was then filtered through a 0.1 pm Nuclepore membrane, washed with a 0.001 mol dm-3 HN03 until the excess Cr3+ and SO$- were removed, and resuspended in 0.001 rnol dnr3 HN03. All chemicals were of analytical reagent grade and were used without further purification. Water was doubly distilled, the second distillation being carried out in an all-Pyrex apparatus.All glassware, except the glass beads and plastic containers, were cleaned with chromic acid. METHODS PARTICLE DEPOSITION AND REMOVAL The packed column (powder-bed) procedure was similar to the kind developed by Clay- field and L ~ m b . ~ The apparatus consisted of a two-piece Pyrex chromatographic column with an i.d. of 9 mm and a reservoir capacity of 50 an3. To support the beads, a stainless steel 170x 700 mesh disc was clamped between the two column sections and sealed by two " 0 " rings. The column reservoir, located above the beads, was adapted to be pressurized slightly by nitrogen gas for flow rate control. The effluent was discharged from the packed bed into a 50 cm3 test tube which was attached to the column with a vent to the atmosphere (fig.1). The glass beads (2.00 g) were fed into the column as a slurry suspended in the same super- natant solution as the Cr(OH)3 sol. This gave a bed height of 1.75 cm once all the beads settled. Next, a measured volume of the chromium hydroxide sol of known number con- centration was forced through the bed. The pH of this sol was adjusted to - 3 which resulted in a stable suspension of positively charged particles (fig. 2). At the same pH the glass beads are negatively charged (table 1) onto which the particles of chromium hydroxide were quantitatively deposited, as ascertained by the absence of a Tyndall beam in the column effluent. The bed was then rinsed with 10 cm3 of the sol supernatant solution to remove any particles not truly adhered.The column was washed free of acid using 20 cm3 of the same solution to which sufficient amount of NaOH was added to raise pH to 8. No particles were desorbed during this washing step.J . E. KOLAKOWSKI AND E. MATIJEVIC 67 Finally, additional volumes (10 and 20 cm3) of the particle-free rinse solution of the same pH (- 8) were passed through the bed, but no particle removal was affected. Thus, one can assume that the particles, deposited as described, did adhere to the glass beads and were not mechanically filtered by the bed. The conclusion was substantiated by the observation that an adjustment of the sol to pH 11.5 prior to passing through the column resulted in no particle adhesion; i.e., the entire amount of chromium hydroxide was recovered in the effluent.FIG. 1.-Packed column used in adsorption and desorption studies. (a) N2 inlet ; (6) column reservoir ; (c) column, 9 m i.d. ; (d) glass beads ; (e) Viton “0 ” rings ; (f) stainless steel 170 x 700 mesh disc ; (g) bleed ; (h) collection tube. To investigate the removal of the adsorbed particles, the bed was washed with 10 cm3 of a rinse solution of a given pH and electrolyte concentration. The procedure was repeated as often as necessary and the particle concentration was determined in each effluent sample. Desorption was always carried out at the same salt concentration as present during particle deposition. Thus, the adsorption and removal steps differed only in the pH of the sol and rinse solutions used. All pH adjustments were made with either HN03 or NaOH.The time of every rinse stage was recorded using a stopwatch and the throughput rates varied between 0.75 and 1.0 cm3 min-l. The rinsing of the column for particle removal was done within an hour following deposition unless otherwise stated. PARTICLE NUMBER CONCENTRATION The particle size distributions were obtained by light scattering using the polarization ratio method.’l* l2 The measurements were carried out with a Brice-Phoenix Model 2000 Universal light scattering photometer. The solid content of the chromium hydroxide stock sol was determined by dry weight analysis for which purpose a deionized sol was kept in a vacuum oven at 75°C until no change in weight was observed. Knowing the average particle size, as obtained by light scattering,68 PARTICLE ADHESION AND REMOVAL and the density of the material (2.42k0.02 g ~ m - ~ ) the particle number concentration of the sol could be readily e~tab1ished.l~ The ratio of scattering intensities, measured at 45 and 0" angles, for vertically polarized light of wavelength 436 nm, was found to be directly pro- portional to the number concentration of the chromium hydroxide sols for dilutions of interest in this work.These optical measurements were, therefore, used to determine the particle number concentration of the effluents in the desorption studies. ELECTROKXNETICS Electrophoretic mobilities of the sol particles were determined with a Rank Brothers Microelectrophoresis Apparatus Mark I1 (Bottisham, Cambridge) using a van Gils cell.Fig. 2 gives the mobilities of the chromium hydroxide particles as a function of pH. Different symbols indicate measurements carried out by four different investigators using different samples of the sols prepared by the same procedure. I I I I I 1 I I 2 4 6 8 10 12 PH FIG. 2.-Electrophoretic mobility of a monodispersed chromium hydroxide sol consisting of spherical particles (modal diameter 0.28 pm) as a function of pH at 25°C. Various symbols show data obtained by four different investigators on two different samples. Zeta potentials, 5, were calculated by means of the Henry equation : c = 1.5vlPeld1 + m a ) ] (1 1 where ,ue is the particle mobility in ,um s-l/V cm-', q the viscosity, E the permittivity of the medium, and the values of the correction term [1+3.(rca)] were taken from tables by Smith.14 In this factor a is the particle radius and K is the reciprocal of the Debye-Huckel distance.Some of the 5 potentials for systems of interest in this work are given in table 1. The zeta potentials of the glass beads were obtained from electro-osmosis measurements carried out with an apparatus described in detail by Mirnik et aE.15 using platinum electrodes, one of which was coated with AgCl. The values of 5 were calculated by means of the expression in which Y is the snecific conductivitv of the solution and h is the done of the nlot obtained 5 = 4 n q ~ b / ~ (2) of the current. for the chromium hydroxide particles. Table 1 gives zeta potentials of glass under the same conditions as reportedJ . E . KOLAKOWSKI A N D E. MATIJEVIC 69 TABLE 1 .-ZETA POTENTIALS OF CHROMIUM HYDROXIDE PARTICLES AND OF GLASS PH 3.1 4.0 5.8 9.6 10.3 11.0 11.5 11.5 11.5 11.5 11.5 11.5 11.5 zeta potential, </mV electrolyte(s) Cr(OH)3 glass HN03 HN03 NaOH NaOH NaOH NaOH HNO, NaOH, lo-, mol dm-3 Ca(N03)2 NaOH, mol dm-, Ca(N03)2 NaOH, lod5 mol dm-3 Ca(N03)2 NaOH, mol dm-3 C~(dipy),(ClO~)~ NaOH, mol dm-3 C~(dipy),(ClO~)~ NaOH, mol dm-3 C~(dipy),(ClO~)~ 46 37 28 - 32 - 37 - 45 - 47 - 23 - 27 - 34 - 19 - 37 - 38 - 36 - 63 - 18 - 48 - 89 - 137 - 43 - 65 - 88 - 43 - 99 - - RESULTS Fig. 3 is a plot of the fraction of chromium hydroxide particles remaining on the glass beads in the column after each of several rinse cycles.Each curve is for a different pH of the wash solutions (adjusted with NaOH).The abscissa is given in time of elution calculated from the known volume of rinse liquid and the flowrate. In all cases approximately the same number of particles was deposited on glass beads (- 1.3 x lolo). Because this number varied from experiment to experiment, data are presented on a relative basis. 1 .o 0.8 i I I I I t 0 2 4 6 8 10 time x 10-3/s FIG. 3 .-Fraction of monodispersed spherical chromium hydroxide particles (modal diameter 0.28 pm) desorbed from glass on repeated elution with rinse solutions of different pH: (0) 9.6, (0) 10.3, (0) 11.0, (0) 12.6 and (A) 11.5. The abscissa is calculated from the known volume of rinse liquid (- 150 cm3) and the flowrate. In all cases, approximately the same number of particles was deposited on glass (- 1.3 x 1O'O) at pH - 3.70 PARTICLE ADHESION AND REMOVAL Although the zero point of charge of the chromium hydroxide particles used is - 8.0, the pH of the rinse solution had to be adjusted to - I0 before any particle desorption occurred.At still higher pH the removal increased, reached a maximum at pH 11.5, and then decreased again when the rinse solution was of pH 12.6. Even at optimum removal conditions a measurable fraction of the particles remained on the beads. The effect of the pH of the sol used in particle deposition on their subsequent removal is illustrated in fig. 4(a). The three curves refer to systems adsorbed at pH 3.0 (as shown before in fig. 3), 4.0 and 5.8, respectively, whereas elution in all cases was carried out with rinse solutions of pH 11.5.Obviously, there was a marked decrease in removal of the particles adsorbed at higher pH values. 1.0 0.8 0.6 0.4 0.2 1 .O 0.8 0.6 0.4 0.2 0 0 - 0 2 4 6 I I 1 I 2 4 6 8 time x 10-3/s RG. 4.-(a) Effect of the pH of the sol during the adsorption stage [( 0) 3.0, (0) 4.0 and (A) 5.8, respectively] on the desorption of chromium hydroxide particles from glass at pH 11.5 for the same system as described in fig. 3. (b) Effect of the ageing period of the chromium hydroxide particles on glass adsorbed at pH 3 prior to rinsing with a solution of pH 11.5 for the same systems as de- scribed in fig. 3. (0) unaged, (0) 21 h, (A) 1 week. The time of ageing after the particles were adsorbed also greatly affected their subsequent separation. Chromium hydroxide was deposited at pH 3.0 and left on the beads, which were in contact with supernatant solution at room temperature for varying periods of time [fig.4(b)]. The column was then repeatedly rinsed with solu- tions of pH 11.5. Ageing for 21 h prior to elution caused a large reduction in de- sorption, and after ageing for 1 week, the removal was negligible. To study the effect of ionic strength and of the nature of the counterions on the desorption of chromium hydroxide particles from glass, several electrolytes in different concentrations were added to the investigated systems. Fig. 5 and 6 show the results obtained when NaN03, NaF, C~I(NO~)~, and Co(dipy)3(C104)3, respec- tively, were present in the rinse solution, the pH of which was always adjusted to 11.5. As described in the experimental section, in each case the adsorption stepJ .E . KOLAKOWSKI AND E. MATIJEVIC (4 (b) 71 0 0 2 4 6 8 time x 10-3/s FIG. S.-Effect of different concentrations of NaN03 (a) and of NaF (b) on desorption of chromium hydroxide from glass for the same system as described in fig. 3. Adsorption pH3.0, desorption pH 11.5. (a) (0) 0.2, (0) 0.1, (A) 0.04, (0) 0.02 and (0) 0.0 mol dm-3; (b) (0) 0.1, (A) O.Ol,!(tl) 0.001 and (0) 0.0 mol dm-3. (4 (6) 1.0 0.8 S 0.6 ' 0.4 z \ 0.2 0 1.0 0.8 0.6 0.4 0.2 A-A-A- I I I 0 2 4 6 0 2 4 6 time x 10-3/s FIG. 6.-Effect of different concentrations of Ca(N03)2 (a) and of Co(dipy)3(C104)3 (b) on desorption of chromium hydroxide from glass for the same system as described in fig. 3. Adsorption pH 3.0, desorption pH 11.5. (a) (A) lW3, (0) lW4, (0) (0) 0.0mol dm-3; (b) (A) (0) (0) (0) 0.0 mol dmd3.72 PARTICLE ADHESION AND REMOVAL was also carried out with chromlum hydroxide sols containing the same concentration of a given salt.The addition of salts can considerably decrease particle desorption and, at suffi- ciently high electrolyte concentrations, the removal can be completely suppressed. The efficiency of a given salt depends strongly on the charge of the counterions. Thus, only mol dm-3 of tris(2,2'-dipyridyl)cobalt(r11)-ion, Co(dipy)i+ , suffices to completely inhibit particle desorption. Calcium ion requires a ten times higher concentration to achieve the same effect (fig. 6), whereas in the presence of 0.2 mol dm-3 NaN03 some removal still occurs (fig. 5). The chelated Co"' ion was used in order to test a highly charged counterion a t high pH.No polyvalent uncomplexed cation is available at pH 11.5. In fig. 5 the two sets of data refer to the addition of NaN03 and NaF, respectively. The latter salt was used to test the possible effect of the fluoride ion. If adsorption of the chromium hydroxide particles on glass was due to chemical bond formation between the silanolic and the =CrOH groups, fluoride ions should greatly affect the removal process. DISCUSSION The rate of particle desorption depends on a number of factors including external forces (e.g., hydrodynamic forces), chemical bonding, particle diffusivity and the interaction potentials between the substrate and the adsorbed material. It is there- fore necessary to first establish which of these parameters may play a role in a given sys tem.The Reynolds number, Re, for the flow through a packed bed is given by l6 where R is the bead radius, G the superficial mass flowrate in g cm-2 s-l (the flow in the absence of the beads), and Po is the bed porosity (void volume per total bed volume). The Reynolds number for the flow rate of 1.0 cm3 min-l, used in this work, was found to be 0.015. This value indicates that hydrodynamics played an insignificant role in the particle desorption. This is further confirmed by the finding that a signifi- cant increase in the rate of flow of the rinse liquid had no effect on the rate of de- sorption. Clayfield and Lumb' showed for their systems that, under similar con- ditions, a variation in flow rates had negligible effect in changing the extent of particle removal.A cursory inspection of the data would seem to indicate that chemical bonding between the acidic silanolic groups of the glass surface and the basic groups on the chromium hydroxide particles may play a significant role in the described adsorption/ desorption phenomena. For example, a sizeable fraction of the particles remains adsorbed even under the most favorable conditions as seen in fig. 3. Furthermore, the higher the pH of the sol at the adsorption step (however, lower than the P.z.c.), the more difficult becomes the particle removal [fig. 4(a)]. One could postulate that the larger number of hydroxyl groups per unit area on the chromium hydroxide particles at higher pH provide more bonding sites for the silanolic groups on glass.Finally, the particle escape decreases with ageing time of adsorbed chromium hydroxide on glass. Considerable effort was made to establish the formation of the =Cr-0-Si= bond between the adsorbent and the adsorbate, but without success. In a series of ex- periments the glass beads were stored overnight at room temperature and at 100°C in 10-l mol dm-3 Cr(NO& solution, the pH of which was adjusted with NaOH to 4, Re = 2RG/y(l -Po) (3)J . E. KOLAKOWSKI AND E. MATIJEVIC 73 Xn neither case could a change in the c-potential of the glass be detected when com- pared to the untreated beads. Under the described conditions, chromium ions are strongly hydrolysed (particularly at the elevated temperature) to give polynuclear cationic complexes. Any adsorption of these species through bonding to silanolic groups would greatly affect the surface charge of glass, ultimately reversing the sign of it.A chromium hydroxide-silanolic condensation bond would also be attacked by fluoride ions, resulting in a much more efficient removal of the adsorbed particles in the presence of NaF than in the presence of NaNO,. Yet with both electrolytes rather similar effects were observed (fig. 5). It would, therefore, appear that the desorption phenomena of chromium hydroxide from glass should be interpreted only in terms of particle diffusion in a region of interacting potentials. In the absence of chemical bonds the interaction between two solids in an ionic medium is due to the contribution of London-van der Waals and electrical double layer potentials.The unretarded attraction potential between a sphere and a flat plate at a distance of separation x < 250 A is 17* l8 x(x+2a) 1 ' A132 x+2a 2a(a+x) 4*(x) = ?Pn - - where A l S 2 is the overall Hamaker constant for the system which can be approximated by A 1 3 2 ( d ~ l - d ~ 3 ) ( d ~ 2 - d ~ 3 ) * (4) sphere-medium-plate ( 5 ) A l l , A22, and A33 are the individual Hamaker constants for interaction between the materials composing the spheres, plates and the aqueous medium (all in vacuo), respectively. The double layer potential between a sphere and a flat plate is given by the follow- ing solution of the linearized Poisson-Boltzman equation in which the upper sign is for the condition of constant surface potentials (t,bl and $ 2 ) and the lower sign for the condition of constant surface charge densities, whereas IC is the Debye-Hiickel reciprocal length.Eqn (6) was derived from the expression of Hogg, Healy and Fuerstenau 2o for the interaction of two dissimilar spheres, taking the radius of one sphere as infinite. The total interaction potential between a sphere and a plate is then given by The potential energies of interaction as a function of separation were calculated for different solutions employed in both deposition and removal of particles in the system chromium hydroxide + glass using the electrokinetic data in table 1 and eqn (4), (6) and (7). In eqn (6) the condition of constant potential was assumed. The overall Hamaker constant, A132 was calculated to be 0.8 kT from eqn (5).This value was based on using = 10.6 kT, for the water-water constant, as ob- tained from the Lifshitz theory and tabulated by Visser.l* For Cr(OH), particles A l = 14.9 kT was determined experimentally from measured coagulation rates of the corresponding sol 21 and for glass, the value A22 = 20.9 kT was calculated by Biittner and Gerlach 22 for the silica-silica interaction. 4 ( 4 = 4 ' 4 ( X ) + 4 R ( X ) . (7)74 PARTICLE ADHESION AND REMOVAL Fig. 7* gives in the insert the total potential energy as a function of distance for adsorption of chromium hydroxide spheres on glass at two different pH values. Obviously, few adsorbed particles are expected to be found many Angstrom units away from the surface due to the fact that 4(x) approaches zero at large separations.The considerable difference in the attraction at two different pH values may be attributed to the thickness of the hydration layer between the substrate and the particles which determines the distance of closest approach, xo. At pH 4.0 the particles penetrate more deeply into the hydration layer and, therefore, are more strongly adsorbed. As a result their removal at higher pH is less efficient as indeed 4c 30 b4 % 2a 3 n -8- 10 0 -I 0 . _ 0 so 100 I50 distance xlA FIG. 7.--Calculated total potential energy curves as a function of distance using the sphere-plate model [eqn (4) and (611 for the chromium hydroxide+glass systems described in fig. 3. Desorption pH : (a) 11.5, (6) 11.0, (c) 10.3 and (d) 9.6. Insert corresponds to systems in iig.4(a) ; adsorption pH : (i) 3.0, (ii) 4.0. observed [fig. 4(a)]. Similar explanation may apply to the ageing effect [fig. 4(b)]. With increasing time of adsorption the particles approach the substrate more closely. Alternately, these effects may be due to surface roughness of the glass beads. On ageing, more particles may move into the surface crevices and thus become more difficult to remove. When solution conditions are changed desorption may occur. In this respect, the pH of the rinse solution plays a major role. Fig. 7 gives total potential energy curves as calculated for the chromium hydroxide+glass system at four different pH values used in the desorption studies (fig. 3). At pH 9.6 attraction still persists * See note added in proof, p. 78.J .E. KQLAKOWSKI AND E . MATIJEVIC 75 and the particles remain adsorbed. Similarly, at pH 10.3 there is no significant potential drop at x > x1 (x, being the location of the maximum) and, therefore, relatively few particles can escape. At higher pH values all particles at sufficient distance xo, which enables them to overcome the energy barrier, will desorb, but will be unable to readsorb. Fig. 8 shows the calculated total potential energy curves for the systems containing Ca2+ and Co(dipy)$+ ions in varying concentrations corresponding to the desorption experiments illustrated in fig. 6. Again these diagrams explain well the observed particle removal phenomena. A comparison of the total potential curves shows that these are quite similar for mol dm-3 Ca2+ and mol dm-3 Co(dipy)z+ and again for mol dm-3 Ca2+ and mol dm-3 Co(dipy)$+. Each of these pairs of salts shows comparable desorption phenomena at cited concentrations (fig.6). Thus, the analogy is striking. 40 - 30 - Q 20- n -8. 3 10- 0- 40 3 0 , 20 10,. .- 0 50 100 IS0 0 100 I so distance x/A FIG. &-Same as fig. 7 for systems described in fig. 6. (a) Ca(NO& : (i) 0, (3) lo-’, (iii) (iii) moi d ~ n - ~ . (iv) lov3 mol dm-3 ; (b) Co(dipy):+ : (i) 0, (ii) The diffusion coefficient, D, for a particle moving normal to a plane surface can be given by where k is the Boltzmann constant, T the absolute temperature and f is the friction coefficient which for a sphere of radius a in a fluid of viscosity q is given by the Stokes law D = kT/f (8) f = 6nqa (9) For the diffusional escape of particles, physically bound to substrates : dN(t) - - I -pN(t) dt76 PARTICLE ADHESION AND REMOVAL where N(t) is the number of particles adhered to the substrate at time t andp is the escape probability. If p is time independent, it becomes a rate constant, and integra- tion of eqn (10) gives : N(t) = N(0) exp [-pi] (1 1) Dahneke 23 gives two solutions for constant p , one describing '' equilibrium " and the other " non-equilibrium " desorption.The equilibrium condition means that for every particle which diffuses away from the substrate and escapes another diffuses toward the substrate and replaces the lost particle. This can happen when there is no potential barrier present ['(x) -+ 01; i.e., when &(x) < 0. To escape the particle must have sufficient thermal energy to leap out of the potential well in order to diffuse away from the substrate.For equilibrium desorption, p is given by where C = [8kT/nrnIt m being the particle mass, and y = 11: exp[ -'(x)/kT] dx. (14) The upper limit of the integral, x l , is the distance at which ' ( x ) reaches a maximum value or, effectively, an asymptotic value. In the case of equilibrium desorption, the maximum tends to zero. The lower limit xo is the distance of closest particle approach to the substrate. Particles within these limits are considered adsorbed and those whose separation exceeds x1 are considered to be "free". The value of '(x,) is taken arbitrarily to be zero so that the potential barrier will be equal to '(x,). In non-equilibrium desorption a greater number of particles diffuse out of the region between xo and x1 than into it.In this case the function 4(x) has a true maximum at xl. Dahneke gives the following expression for the probability of particle escape in the non-equilibrium situation This expression is valid if 4(x) can be represented as a parabola in the vicinity its maximum as follows : 5 ) of where x x1 and w2 is the 2nd derivative or curvature of ' ( x ) evaluated at its maximum. Eqn (15) was found to be valid for +(xl) 2 10 kT.24 This means that particle desorption has to occur at a slow enough rate such that the distribution of particle energies remains unchanged. The non-equilibrium solution does not take into account particle readsorption into a secondary minimum of ' ( x ) .If the peak of 4(x) is sharp enough such that mo21f2 $ 1 then the factor in brackets in eqn (15) approaches unity and eqn (15) reduces to eqn (12), the equilibrium solution. The physical interpretation of particle escape over a sharp maximum is that the particles have so short a distance x to travel in order to be free of the substrate that friction has a negligible role in impeding their escape.J . E . KOLAKOWSKI AND E . MATIJEVIC 77 The function -In [N(t)/N(O)] was plotted against time for all desorption condi- tions studied. In several cases the curves were linear indicating that p is time inde- pendent [fig. 91. Using the experimentally determined rate constants in eqn (ll), xo was determined for four of the five desorption conditions. For this purpose, different values for xo were substituted into the following expression : which is a rearranged eqn (15), until the root of F(xo) was found.Eqn (9) was used to calculate the coefficient$ The results are given in table 2. The above method cannot be used to calculate xo for desorption data at pH 9.6, since the function 4(x) at its peak cannot be represented by a parabola. 1.5 1.0 ;;2 \ m c1 z u a 0.5 I 0 2 4 6 8 10 time x lO-”/s FIG. 9.-Plot according to eqn (11) of the data obtained for the desorption of chromium hydroxide spherical particles from glass under several different conditions illustrated in fig. 3 and 4. Adsorption pH: (V) 3.0 (21 h ageing), (A) 3.0 (1 wk ageing), (0) 3.0, (0) 3.0, (0) 5.8; desorption pH: (V) 11.0, (A) 11.0, (0) 9.6, (0) 11.0, (0) 11.5.It is also recognized that eqn (9) is an approximation. Calculations, based on expressions for the frictional coefficient, which consider particles moving normal to a planar surface,23 have shown that the error in using eqn (9) does not substantially affect the results given in table 2. TABLE 2.-vARIOUS PARAMETERS CALCULATED FOR THE DESORPTION OF CHROMIUM HYDROXIDE PARTICLES FROM GLASS desorption conditions PH desorption p x 1051s-1 ~01.4 (b(x1)lkT PH adsorption 3.0 11.0 13.7 6.3 26.5 3.0 9.6 1.1 3.0, aged 1 w 11.5 1 .o 9.1 29.4 3.0, aged 21 h 11.5 9.1 9.5 27.4 5.8 11.5 9.8 9.5 27.4 - -78 PARTICLE ADHESION A N D REMOVAL It is reasonable to expect that the adsorbed particles will populate the region of lowest energy, i.e., at the distance xo from the substrate.However, the high energy barriers which correspond to the values of xo (table 2) indicate that the modal particle separation cannot be at xo and that the hydration layer on the glass and/or on the particles must greatly affect the particle separation distance. Otherwise no desorption would be likely to occur, owing to the high energy barriers. The apparent in- consistency is probably due to errors in the calculation of the 4(x), which arise from the uncertainty in the value of the Hamaker constants and the application of the linearized solution of the Poisson-Boltzmann equation, eqn (6), beyond its range of validity (Le., $ < 25 mv). Both of these factors can greatly alter the 4(x) against x curves. Finally, calculations based on the constant charge model gave much higher values of #(x).Note added in proof.- The total potential energies (+/kT) as plotted in fig. 7 and 8 are about one order of magnitude too low. This error was introduced in the calculation of the double layer repulsion energies +R(x) [eqn (6)] which were underestimated by a factor of 4n. The latter factor must be introduced when S.I. units are used, Fortunately, this error in calculation does not affect any of the conclusions as derived from the particle desorption studies. E. M. is indebted to Drs. J. G. Maroto and M. A. Blesa of CNEA, Buenos Aires, Argentina, for pointing out the erroneous magnitude of the total potential energy values. The authors are indebted to Profs. Eric Clayfield (Shell Research Centre, Thornton, England), Barton Dahneke (University of Rochester) and Eytan Barouch (Clarkson College) for many valuable discussions.The assistance of Mr. Terry A. Ring in various computations is greatly appreciated. We acknowledge the helpful remarks of one referee. This work was supported by a grant from N.S.F. V. Levich, Acta Physicochim. U.S.S.R., 1942, 17, 257. A. C. Riddiford, Adv. Electrochem. Electrochem. Eng., 1966,4,47. J . K. Marshall and J. A. Kitchener, J. Colloid Interface Sci., 1966, 22, 342. M. Hull and J. A. Kitchener, Trans. Faraday Soc., 1969, 65, 3093. ’ G. E. Clint, J. H. Clint, J. M. Corkill and T. Walker, J. Colloid Interface Sci., 1973, 44, 121. P. H. Tewari and A. B. Campbell, Abstracts 172 A.C.S. Nat. 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